Lagrangian correlation coefficient

We assume that the Lagrangian correlation coefficient for fluid motion may be represented by an exponential function... 

Taylor eddy diffusion coefiicieni for the turbiijeni iliiid (Goldstein. 1938, p. 217). However, the correlation coefficient in (4.54) applies to the gas velocities over the path of the particle. Heavy particles move slowly and cannot follow the fluid eddies that surge around them. Thus the time scale that should be employed in (4,54) ranges between the Lagrangian scale for small particles that follow the gas and the Eulerian lime scale for heavy particles that remain almost fixed (Fricdlander, 1957),... 

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

Similar transition expectation values can also be defined for other non-variational methods like Mqller-Plesset perturbation theory, where one defines a Lagrangian by adding the equations for the correlation coefficients as extra conditions multiplied with Lagrangian multipliers to the respective MP energy expression (Hattig and Hefi, 1995 Aiga and Itoh, 1996). 

The first term is the generalization of the normal MP2 energy, Eq. (9.68), to the case of time-dependent molecular orbitals and time-dependent first-order doubles correlation coefficients i2 [l](i)- The second and third terms are the time-dependent version of the equations for the [1] (t) coefficients multiplied with their Lagrangian multipliers... 

Eq. (11.32). The last term is therefore the TDHF equations multipfied by corresponding time-dependent Lagrangian multipliers Tpq t). This Lagrangian is thus variational with respect to the TDHF coefficients, Eq. (11.35), the first-order doubles correlation coefficients as well as with respect to the Lagrangian multipliers for both types of coefficients. A constraint like Eq. (12.23) is not necessary in the QED method as a result of the fact that the second derivative is with respect to Ta... —ijj). The TDHF coefficients have to be obtained by solving the TDHF equations, Eq. (11.35), as in the FED method. The first-order MP2 amplitudes as well as the Lagrangian multipliers for both the TDHF coefficients and the first-order MP2 amplitudes are obtained by solving appropriate response equations. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

In addition, it is dubious whether this new correlation due to Brucato et al. (1998) should be used in any Euler-Lagrangian approach and in LES which take at least part of the effect of the turbulence on the particle motion into account in a different way. So far, the LES due to Derksen (2003, 2006a) did not need a modified particle drag coefficient to attain agreement with experimental data. Anyhow, the need of modifying particle drag coefficient in some way illustrates the shortcomings of the current RANS-based two-fluid approach of two-phase flow in stirred vessels. 

This is the approximation of a Wiener variable, which renders the system (4.3) a coupled Langevin system. The correlation function for F is therefore of the character of a diffusion coefficient, and the system can be solved by standard Lagrangian methods. 

The quantity A can be regarded as a functional of the correlation functions has, Cas and tas, as well as the MO coefficient vh and the Cl coefficient Ci. Imposing the constrains to the orthnormality of configuration state functions and one particle orbital, we have defined the following Lagrangian,... 

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